Arithmetica

1. A unit is that according to which each existing thing is said to be one; therefore, here, the unit is the first measure of discrete quantity.
2. A number is a multitude composed of units.
3. A derived number is a specific numeric operation defined by a number (or the unit) and numerical operations.

1. For any multitude, one and only one number corresponding to that multitude can be composed by the addition of units to each other.
2. For any number, N, it is always possible to compose another number, M, by the addition of the unit to N.

   Cor. 1 and Definition 4: If the unit is added to N only once, then M is said to be the successor to N (or the smallest number that is larger than N); we denote the successor of a number N by N’ (read “the successor of N” or “N prime”).

   Cor. 2: Two numbers cannot have the same successor.

   Cor. 3: The unit is the successor to no number.

3. To prove that (quia) something is a property of every number as such, the property can be proved true of the unit and then of the successor of some unspecified number (i.e. of the successor of any given multitude composed of units); such a proof proceeds from the properties of the unit and of numbers as such, and proof is possible in this way because numbers can be produced infinitely with addition of the unit. Here, we consider one as a number only analogously; the unit may be said to be numeric because it is the principle of numbers.

1. Things equal to the same thing are also equal to one another. (Equal is symbolized = and unequal ≠, as usual.)
2. If equal things are added to equal things then the wholes are equal. (Addition is symbolized +.)
3. If equal things are subtracted from equal things then the remainders are equal. (Subtraction is symbolized -.)
4. The whole is greater than the part.

Capital letters from the beginning of the English alphabet stand for one or any given number, A, B, C, D, E, and F. Arabic numerals in the usual way stand for the one and the numbers (1, 2, 3, 4, etc.). Parentheses group parts of an expression to show the order of evaluation of the parts of the expression, with the innermost set of parentheses being evaluated first.

The goal is to lay out in detail how to construct and prove truths about the numbers as such and the derived numbers, which are specific numeric operations. This occurs through consideration of quantity as such, and also through consideration of how quantitative operations relate to each other and to numbers.

1. Numbers as such and derived numbers. A number as such is a multitude measured by the unit. The unit is the principle of number and is treated as numeric by analogy (Axiom 3). Whatever else is called a number in this work is a derived number: a specific numeric operation, or that by which such an operation is defined. The positive and negative numbers, zero, the parts called by the name of a number, the ratios, the bounds, the real numbers, the complex numbers and i are all of this kind. None of them is a number as such.
2. How an operation belongs to arithmetic. A derived number is not considered as an action in some other genus. It is considered as a relation among quantities, specified wholly by quantity: adding three has no content beyond three and the relation of the sum to the operand. So the derived numbers belong to arithmetic as what is said of numbers, and the subject remains number.
3. Sameness where both apply. Nothing is created and nothing is discarded. When a later kind answers to an earlier one, as a whole ratio to a number, a rational bound to a ratio, or a real number to a bound, the earlier remains what it was, and the two are the same where both apply.
4. Any, not all. No completed infinite multitude is supposed. A statement about every number, ratio or bound is a statement about any one that may be taken. A rule may settle something for any ratio; the ratios are never gathered into a whole.
5. Determinate is enough. A question that is determinate has an answer, whether or not a finite procedure finds it.
6. Function. A function is a rule that takes an operand and yields a result. Rules that differ are different functions, even when they yield the same for every operand; such rules are equivalent.
7. Proper proof. Where a truth holds of numbers and also of other things, the proof given is the one proper to arithmetic and not a common proof.
8. Completion of the operations. Each kind of derived number is admitted because with it an operation can be undone that could not be undone before: removing, dividing, taking the side. i is admitted on this ground. With it the numeric operations are complete, in the sense of Definition 91. Theorem 434 proves this for the taking of a side. Theorem 435 states it in full, and its proof is not given in this work.

The numbers in parentheses after a theorem or definition name its source: Euclid's Elements by book and proposition, or E. Landau's Foundations of Analysis by his numbering. The enunciations taken from Euclid follow Heath's translation. For the arithmetical books the proofs are Euclid's own, in his lettering, abridged; where a proof supposes something Euclid does not state, a note names the theorem that supplies it. For what Euclid proves of magnitudes, the enunciation is his and the proof is drawn from what a bound is. Landau's theorems are proved after his pattern, without his sets: a class of fractions, a cut and a division of all real numbers are each replaced by what the Specifications allow.